Why don't I see light reflected off a mirror in Cycles?
Lights are only visible in Cycles when they are mesh lights. If you want a light source to be reflected you need to make it from geometry and create an emission shader for it.

If you wish light to be reflected and illuminate other objects, you need to enable reflective caustics:



These calculations take a lot of time in Cycles and are inefficient, because of the way tehy are calculated. For better caustics one would need to use a render engine that supports bidirectional pathtracing. I think LuxCoreRender can do a better job with this.